Application of the impedance measurement technique for demonstration of an adiabatic quantum algorithm.

1
Application of the impedance measurement
technique for demonstration of an adiabatic
quantum algorithm.
M. Grajcar, Institute for Physical High
Technology, P.O. Box 100239, D-07702 Jena,
Germany and Department of Solid State Physics,
Comenius University,SK-842 48 Bratislava,
Slovakia A. Izmalkov, E. Ilichev, H.-G.
Meyer Institute for Physical High Technology,
P.O. Box 100239, D-07702 Jena, Germany We
are grateful to our coauthors N. Oukhanski, D.
Born, U. Hübner, T. May, Th. Wagner, I.
Zhylaev, Ya. S. Greenberg, H. E. Hoenig, W.
Krech, M. H. S. Amin, A. Smirnov, Alec Maassen
van den Brink, A. M. Zagoskin for their help and
contribution to this work on different stages.
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Contents

• Introduction
• A. Adiabatic quantum computation
• B. Measurements by parametric transducer
• Control of the evolution for
• A. Single qubit
• a. Classical regime.
• b. Landau-Zener transitions.
• c. Adiabatic response
• B. Two coupled qubits adiabatic behaviour
• C. Three coupled qubits - MAXCUT problem
• Readout of the adiabatic quantum computing for 3
  coupled qubits
• Conclusions

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Adiabatic quantum computing
There are a set of problems, which can be solved
by quantum algorithm more efficiently than by
conventional one.

1. Start with initial Hamiltonian HI with known
   ground state Igt
2. Make adiabatic evolution from Igt to the unknown
   ground state ggt of HP
3. Readout the ground state of HP

Realization for superconducting flux qubits
1) For flux qubits we choose initial Hamiltonian
HI with trivial ground state 0gt
2) Changing the bias of individual qubits
adiabatically, Hamiltonian Hi is transformed to
HP.
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Parametric transducer

• Mechanical displacement

?x ? ?? ?x 6 10-17 cm
Braginski et. al., JETP Lett., 33, 404, 1982.
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Resonator

• Quantum nondemolution measurements of a
  resonators energy

??
?
???L/v(E)
Dielectric sucseptibility

1. No perturbation of the measured observable
2. The canonically conjugate to the measured
   observable is perturbed according to uncertainty
   principle.

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Experimental method
?i?e?LI(?i) L?0 F?i ??e
A Golubov, M. Kupriyanov and E. Ilichev, Rev.
Mod. Phys., to be publ in April, 2004 E. Ilichev
et al. Rev. Sci. Instr., 72., 1883, 2001 E.
Ilichev et al. Cond-mat 0402559
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Experimental setup
Advatages 1) high quality PRC - narrow
bandwidth filter 2) high sensitivity Q 103 3)
small coupling coefficient k10-2 ? small
back-aktion of the amplifier
HEMT
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Classical and Quantum regime
2
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Nb persistent current qubit in classical regime
T800 mK
T500 mK
T100 mK
Al-qubit
T20 mK
Junction area 3x3 mm2 EJ/Ec?104
E. Ilichev et al., APL 80 (2002) 4184
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Al persistent current qubit placed in Nb coil

• Nb coil is prepared on oxidized Si substrates by
  optical lithography.
• The line width of the coil windings was 2 ?m,
  with a 2 ?m spacing.
• Various square-shape coils with between 20 and
  150 ?m windings were designed.
• We use an external capacitance CT.

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Al persistent current qubit

• Material Aluminium, Shadow-evaporation tecnique
• Two contacts 600x200nm, (IC ? 600 nA), the third
  is smaller, so that aEJ1 /EJ2,3 0.8-0.9
• Inductance L ? 20 pH
• J.E. Mooij et al., Science 285, 1036, 1999.

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Idea of measurements Landau-Zener tunneling
energy at different external fluxes
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Idea of measurements Landau-Zener tunneling
External flux ?rf ?dc
?dc
energy at different external fluxes
V
?dc
Voltage across the tank vs ?dc
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Idea of measurements Landau-Zener tunneling
External flux ?rf ?dc
?dc
energy at different external fluxes
V
?dc
Voltage across the tank vs ?dc
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Tank voltage vs external flux near f0.5
Tmix. Chamber 10 mK
A. Izmalkov et al., EPL, 65, 844, 2004
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Classical and Quantum regime
2
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Phase shift ? vs F near degeneracy point f0.5
Ya. S. Greenberg et al., PRB 66, 214525, 2002 M.
Grajcar et al., PRB 69, 060501(R), 2004
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Phase shift ? vs F at different temperatures
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Temperature dependence of the dip height and width
?T3
Th200 mK
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Two coupled flux qubits
Idc2
Mab
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Determination of device parameters
T160 mK 90 mK 50 mK 10 mK
T160 mK 90 mK 50 mK
IMT deficit
Theory
Experiment
Two-qubit Hamiltonian

• the width of one-qubit dips gives Da450 MHz,
  Db550 MHz
• The height of one-qubit dips gives persistent
  currents IaIb 320 nA and J MabIaIb 410 MHz.

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Temperature dependence of the IMT dip amplitudes
The measure of entanglement is Concurrence C For
pure state
The concurrences of eigenstates of our two-qubit
system are C1C40.39, C2C30.97

• We substitute Da, Db Ia, Ib, J, determined from
  Low-T measurements. The T-dependence of IMT dips
  amplitudes agrees with these values.
• Ratio of the dips grows with T, because the
  thermal excitations tend to destroy coherent
  correlations between the qubits.

But the equilibrium concurrence Ceq (10
mK)0.33 In our experiment the minimal
temperature of the sample was about 40 mK, where
Ceq0.
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MAXCUT problem

• The MAXCUT problem is part of the core
  NP-complete problems
• MAXCUT adiabatic quantum algorithm already
  demonstrated by NMR
• M. Stephen et al., quant-ph/0302057

Simple example for 4 nodes
S30
S40
w3
w34
w4
w24
w23
w2
0
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w14
w13
1
w12
w1
S11
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Hamiltonian of N inductively coupled flux qubits
Payoff function is encoded in Hamiltonian HP if
?iltltJi,j and
HP The MAXCUT problem Hamiltonian
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Readout by parametric transducer
0gt
1gt
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First three energy levels of the three qubit
system during readout
Parameters
sgt 000 001 010 011 100 101 110 111
E(K) -0.034 -0.473 -0.599 -0.054 0.185 -0.019 0.107 0.887
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Measured quantity d2E/df2
S21
S10
S30
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Conclusions

1. By making use of the Parametric Transducer
   measurements the flux qubits can be completely
   characterized
2. Adiabatic quantum computing is operated even if
   the system is in mixed entangled states.
3. Parametric Transducer can effectively readout the
   result of the adiabatic quantum computing leaving
   the system in the ground state during and after
   the measurement. Such non-demolition
   measurement naturally follows the idea of
   adiabatic quantum computing.